Midterm1_Solutions

Midterm1_Solutions - J M Mendel Spring 2007 EE 364 Midterm 1 Closed Book One 81/2”Xl 1” sheet of notes(both sides permitted 1(20 points A die

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Unformatted text preview: J. M. Mendel Spring 2007 EE 364 Midterm # 1 Closed Book One 81/2”Xl 1” sheet of notes (both sides) permitted 1. (20 points) A die is tossed twice and the number of dots facing up is counted and noted in the order of occurrence. Let A be the event “total number of dots is even,” and let B be the event “both tosses had an even number of dots.” Find: a. (4 points) P(A) b. (4 points) P(B) c. (6 points) P(A l B) d. (6 points) P(B | A) 2. (30 points) For the network below all unlabeled links never fail. The probability that links A, B, C, D transmit are a, b, c, d, respectively. All links behave independently. a. ( 10 points) Let T12 denote the transmission between points 1 and 2. Find an expression, in terms of mutually exclusive events, for T12 using sets. b. (10 points) What is the probability of transmission between points 1 and 2? c. (10 points) Suppose that a transmission exists between points 1 and 2, what is the probability that B is transmitting? 3. (25 points) A dart is thrown onto the square shown below. Assume that the dart is equally likely to fall anywhere in the square. Let the random variable Z be given by the sum of the two coordinates of the point where the dart lands. a. (3 points) Describe the sample space of Z, 52. b. (8 points) Sketch the region in the square corresponding to the event {Z S 2} for —oo < z < oo. [Hintz there are two cases] To receive full credit your diagrams must be completely labeled, and you have to quantify the two cases. c. (10 points) Find P(Z S z). [Hint Use geometry] d. (4 points) Find the density function f(z) . J. M. Mendel Spring 2007 y 4. (25 points) Let X and Y have joint density (see the diagram below) 3/ 2 in Region A f (x,y) = . . 1/ 2 1n Region B )J 1 x 0 1 Find [Hint For Parts a—c, use geometry]: :1. (5 points) P(X<1/2,Y >1/2) b. (5 points) P(X <1/2IY > 1/2) c. (8 points) P(min(X,Y) < 2 / 3) d. (6 points) fX(x) @3le MM¥H engag- (Q) A:<(l,l)’(l,3)l(l,$), (2,13)(2’L+))(2’g)1 , "7.7) (6,2)’(G,<Q)'(Qle)y , Arhax’). 6K73=IEW 7 7 8:{C2‘z)2 (life); (2’6), (LF’ I (ql‘d, (*16) (¢.L),(c,%!,(e,(.)§ / E» 4% 4M x (12) 19(6): 9/36; r ([4. 6) Male): PM“ 63 2 ‘PBS 2, W5) Me) Noitjcwvd: 906: 6 W MA- (cm my») 2 #4063 2 We) 2 ’{4 4/2 _——__.. a PM) 19(9) '12 (b) ’PCM): P04 U 9'6: u A'GC'D) : a +O~43bc —t—((~a.)b((—c) A : Q+b6~qbc+bd~ECcl~abd+qloccfl :: q +Ec+bdsabc~fgcd-algd+abc4 (C) Pash-sz) = P(TI>—JB) We) P(T.7_) WM ’PCMIQmam aw. Gum Mm“- W‘)Mzm€c¢p¢m W‘.¢(mm W .‘ pr) A“: M). m'b "Muce/ 1708 'I’nfik ?( AUA'CLUA‘c'D) $(9)+P(A‘o)+pm‘c‘t>7 a4. ((—m)c+(t—d(‘-CH a +c+d ~QC—ad-cd + acé: r n H H “ewe b+cL ME- foc—-«lod-- lo d-Hdoaé 7) 6 7/: a a C ( h!» a+bc+b&—¢EC~ECA “Qbé +5?ng CW2: b<§429' Nfl:%%g=3~ng-:Lad 7K=$Htajmoel b—(zJa) =2L—3. mm. , O “6&0 “$5.241 é“? osasg |_%(29~$)7§ 55322}, ‘ 532B Nomrrweammmm :r A C\_ (mofmw 5/5? W255): l~ ‘P(%>;) Uwe, ...
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This note was uploaded on 02/27/2008 for the course EE 364 taught by Professor Mendel during the Spring '08 term at USC.

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Midterm1_Solutions - J M Mendel Spring 2007 EE 364 Midterm 1 Closed Book One 81/2”Xl 1” sheet of notes(both sides permitted 1(20 points A die

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